Hi @scotti,
We have this problem all the time with our own workshops, so I totally feel your pain.
RStudio Package Manager is a server product, so you'd need one license and could serve packages to your whole class. That said, if the step you're getting hung up on is installing system packages (like Tex) on individual student laptops, RStudio Package Manager isn't going to help that much.
One thing I'd definitely recommend is having your students install Tex via the TinyTex R package if you're not already. It's definitely the easiest way to get Tex up and running for use with RMarkdown.
Another option would be to have the students use rstudio.cloud. RStudio.cloud is an online server environment where you could provision classrooms for your students to work. It would let them get up and running in an RStudio session by just logging in with credentials you'd provide. RStudio.cloud is currently free as it's still in its alpha version.
Hope that's helpful!